Page 44 - Determinants and Their Applications in Mathematical Physics

3.3 The Laplace Expansion 29

The expansion formula (3.3.4) follows.

Illustrations
1. When r = 2, the Laplace expansion formula can be proved as follows:
Changing the notation in the second line of (3.3.10),

n n

A = a ip a jq A ij;pq , i < j.
p=1 q=1
2
This double sum contains n terms, but the n terms in which q = p are
zero by the definition of a second cofactor. Hence,

A = a ip a jq A ij,pq + a ip a jq A ij;pq .
p<q q<p
In the second double sum, interchange the dummies p and q and refer
once again to the definition of a second cofactor:

A = a ip a iq
A ij;pq
p<q a jp a jq

= N ij;pq A ij;pq , i < j,
p<q
which proves the Laplace expansion formula from rows i and j. When
(n, i, j)=(4, 1, 2), this formula becomes

A = N 12,12 A 12,12 + N 12,13 A 12,13 + N 12,14 A 12,14
+ N 12,23 A 12,23 + N 12,24 A 12,24
+ N 12,34 A 12,34 .
2. When r = 3, begin with the formula

n n n

A = a ip a jq a kr A ijk,pqr , i<j <k,
p=1 q=1 r=1
which is obtained from the second line of (3.3.11) with a change in
3
notation. The triple sum contains n terms, but those in which p, q,
and r are not distinct are zero. Those which remain can be divided into
3! = 6 groups according to the relative magnitudes of p, q, and r:

A = + + + + +
p<q<r p<r<q q<r<p q<p<r r<p<q r<q<p
a ip a jq a kr A ijk,pqr .